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Existence of almost periodic solutions to some stochastic differential equations. (English) Zbl 1130.34033
The concept of $p$th-mean almost periodicity for Banach-space-valued stochastic processes is studied. Some preliminary results are applied to verify existence and uniqueness of mean-square almost periodic mild solutions for semilinear stochastic evolution equations $$ dX(t) = [ A X(t) + F(t,X(t)) ] dt + G(t,X(t)) dW(t) $$ with mean-square periodic, Lipschitz-continuous nonlinearity $F$ and driven by Brownian motion $W$. For this purpose, they make use of well-known Banach’s fixed point principle.

34F05ODE with randomness
34C27Almost and pseudo-almost periodic solutions of ODE
35B15Almost and pseudo-almost periodic solutions of PDE
37L55Infinite-dimensional random dynamical systems; stochastic equations
60H10Stochastic ordinary differential equations
60H15Stochastic partial differential equations
60H20Stochastic integral equations
60H25Random operators and equations
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